Question

If h(x) = (f o g)(x) and h(x) = (5/x + 1)², find f(x) and g(x).

A) f(x) = 5/x, g(x) = 1
B) f(x) = 5/x², g(x) = x + 1
C) f(x) = 5, g(x) = x + 1
D) f(x) = 5x, g(x) = 1/x + 1

Answer 1

To find f(x) and g(x) such that h(x) = (f o g)(x) = (5/x + 1)^2, we determine that f(x) must square the result of function g(x). The correct pair of functions that fulfill this is f(x) = 5/x² and g(x) = x + 1, corresponding to Option B.

Step-by-step explanation:

If we have the composition of two functions h(x) = (f o g)(x) and it's given that h(x) = (5/x + 1)², we're looking to find two functions f(x) and g(x) such that f(g(x)) will give us h(x). The operation indicated by (f o g)(x) means that g(x) is the inner function and f(x) is the outer function that is applied to the result of g(x).

To determine f(x) and g(x), we can deduce that since h(x) is squared, g(x) must be giving us something that looks like (5/x + 1), and then f(x) must be squaring this result to match h(x). If we look at the options:

• Option A: Can be quickly dismissed because it doesn't include squaring.
• Option B: Is correct. f(x) = 5/x² will square the function g(x) = x + 1.
• Option C: Is incorrect, as f(x) here is not squaring the result of g(x).
• Option D: Is incorrect because it introduces a multiplication not present in h(x).

Therefore, the correct functions f(x) and g(x) are f(x) = 5/x² and g(x) = x + 1, which corresponds to Option B.